Mathematics

ECMM743 - Logic, Models and Sets (2018)

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MODULE TITLELogic, Models and Sets CREDIT VALUE15
MODULE CODEECMM743 MODULE CONVENERDr Robert Barham (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
Number of Students Taking Module (anticipated) 15
DESCRIPTION - summary of the module content

The goal of this module is to apply the power of mathematical reasoning to itself, and to properly examine the primary tool of Mathematics, namely logic.  This study is motivated by three things.  First, its own intrinsic interest.  Secondly, its applications to other branches of Mathematics.  We won't be able to get to any of the direct applications of logic in one term, but this course will develop your proof skills along with other indirect applications.  Finally, what logic can tell us about the nature of Mathematics.   Many philosophical questions (such as ``Could all mathematics be done by a computer?'') cannot be properly answered without understanding this material.

This module will start by introducing propositional logic and deduction in propositional logic, as well as examining some of its properties (such as completeness and soundness).  Then we will do the same for predicate calculus, and then go on to discuss some of the basics of Model Theory, showing how predicate calculus can describe mathematical structures.  We will then examine the Zermelo-Fraenkel axioms of Set Theory, and study the theory of ordinals and cardinals.

You will be expected to have taken the prerequisite ECM2711 module Groups, Rings and Fields in the second year, but a perfect recollection will not be necessary.

 

AIMS - intentions of the module

This module aims to give you a foundation in the study of logic, and to use that to provide an introduction to two of the branches of Logic, namely Model Theory and Set Theory.  Through this course, you will become fluent in formal logical reasoning, be able to formalise mathematical structures as first order models, and clearly and rigorously answer questions about infinity.

 

 

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

1.  perform formal proofs in both propositional logic and predicate calculus;

2.  formalise mathematical structures into first order logic;

3.  describe the properties of a first order theory;

4.  perform cardinal and ordinal transfinite arithmetic;

5.  prove statements about any of the systems mentioned in 1.-4.

Discipline Specific Skills and Knowledge

6.  state and apply definitions in the field of logic and applications;

7.  write informal proofs and translate these into formal arguments;

Personal and Key Transferable / Employment Skills and Knowledge

8.  reason logically and abstractly;

9.  communicate complicated ideas in writing, professionally and using correct mathematical notation.

 

SYLLABUS PLAN - summary of the structure and academic content of the module

- Syntax of propositional logic. Deduction in propositional logic.  Properties of propositional logic.

- Syntax of predicate calculus.  Definition of model.  Deduction in predicate calculus.  Properties of predicate calculus.

- Definition of models, theories and axiomatisation.  Definition of decidability and completeness.  The Compactness Theorem.

- Introduction of ZFC.  Cardinal and Ordinal arithmetic.
 

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 33.00 Guided Independent Study 117.00 Placement / Study Abroad 0.00
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities 28 Lectures
Scheduled learning and teaching activities 5 Tutorials
Guided independent study 12 Formative assignments
Guided independent study 20 Coursework assignments
Guided independent study 85 Lecture and examination preparation; wider reading

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Four Exercise Sheets 3 hours each All  Written and Verbal Comments
       
       
       
       

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Exam mark. Verbal/written feedback
Coursework 10 10 hours All Mark, written comments and mark scheme.
Coursework 10 10 hours All Mark, written comments and mark scheme.
         
         

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
All Above Written Exam (100%) All August Ref/Def Period
       
       

 

RE-ASSESSMENT NOTES

Referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Basic reading:

 

ELE: http://vle.exeter.ac.uk/

 

Web based and Electronic Resources:

 

Other Resources:

 

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Chiswell I and Hodges W Mathematical Logic, Oxford Texts in Logic 3 Oxford University Press 2006 [Library]
Set Cori R. and Lascar D Mathematical Logic, Part 1 Oxford University Press 2000 9780198500490 [Library]
Set Cori R. and Lascar D. Mathematical Logic, Part 2 Oxford University Press 2000 9780198500506 [Library]
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES ECM2711
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 7 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Thursday 06 July 2017 LAST REVISION DATE Thursday 28 February 2019
KEY WORDS SEARCH Mathematical logic; Model Theory; Set Theory; Propositional Logic; Predicate Logic